2 related rate problems I can't solve (cross-posted to math-help)
I have 2 related rate problems I can't solve. Any suggestions, hints, help are appreciated!
First problem: A highway patrol airplane flies 3 miles above a level, straight road at a constant rate of 120 mph. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 5 miles the line-of-sight distance is decreasing at a rate of 160 mph. Find the car's speed along the highway.
What I've done so far:
Here's how I've labeled the given information
Y=3 (vertical distance from the road to the plane)
dx/dt=120 mph (rate of change of the plane)
s=5 (distance from plane to car)
ds/dt=-160mph (rate of change from plane to car)
I need to find the rate of change (speed) of the car.
What I can't figure out: What's an equation I can use to relate these rates? Once I have that I should be able to differentiate with respect to time and plug in my known values to find the speed of the car. I just can't figure out how to relate these variables!
Second problem: A man 6 feet tall walks at a rate of 5 ft/sec toward a streetlight that is 16 feet above the ground. A ball is dropped from the same height from a point 30 feet away from the light. How fast is the ball's shadow moving along the ground 1/2 second later? (Assume the ball falls a distance s=16t^2 in t seconds.)
What I've done so far:
I've labeled the given information as follows:
y=50 (height of pole)
x(t) is the rate of change of the shadow, which is what I need to solve for.
Then I plugged t=1/2 into the equation s=16t^2 to get s(1/2)=4 feet. Then I subtracted 4 feet from 50 feet to get 46 feet, the height of the ball after 1/2 second.
I'm assuming that the related rate equation has something to do with similar triangles....like 50 is to ? as 46 is to ? but that's as far as I got.
Thanks in advance for the help!
First problem: A highway patrol airplane flies 3 miles above a level, straight road at a constant rate of 120 mph. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 5 miles the line-of-sight distance is decreasing at a rate of 160 mph. Find the car's speed along the highway.
What I've done so far:
Here's how I've labeled the given information
Y=3 (vertical distance from the road to the plane)
dx/dt=120 mph (rate of change of the plane)
s=5 (distance from plane to car)
ds/dt=-160mph (rate of change from plane to car)
I need to find the rate of change (speed) of the car.
What I can't figure out: What's an equation I can use to relate these rates? Once I have that I should be able to differentiate with respect to time and plug in my known values to find the speed of the car. I just can't figure out how to relate these variables!
Second problem: A man 6 feet tall walks at a rate of 5 ft/sec toward a streetlight that is 16 feet above the ground. A ball is dropped from the same height from a point 30 feet away from the light. How fast is the ball's shadow moving along the ground 1/2 second later? (Assume the ball falls a distance s=16t^2 in t seconds.)
What I've done so far:
I've labeled the given information as follows:
y=50 (height of pole)
x(t) is the rate of change of the shadow, which is what I need to solve for.
Then I plugged t=1/2 into the equation s=16t^2 to get s(1/2)=4 feet. Then I subtracted 4 feet from 50 feet to get 46 feet, the height of the ball after 1/2 second.
I'm assuming that the related rate equation has something to do with similar triangles....like 50 is to ? as 46 is to ? but that's as far as I got.
Thanks in advance for the help!